U.S. patent application number 14/658408 was filed with the patent office on 2015-07-02 for analyzing device.
This patent application is currently assigned to SUMITOMO HEAVY INDUSTRIES, LTD.. The applicant listed for this patent is SUMITOMO HEAVY INDUSTRIES, LTD.. Invention is credited to Daiji ICHISHIMA, Yoshitaka OHNISHI.
Application Number | 20150186573 14/658408 |
Document ID | / |
Family ID | 50340835 |
Filed Date | 2015-07-02 |
United States Patent
Application |
20150186573 |
Kind Code |
A1 |
OHNISHI; Yoshitaka ; et
al. |
July 2, 2015 |
ANALYZING DEVICE
Abstract
An analyzing device analyzes a particle system defined in a
virtual system by numerically calculating a governing equation that
governs motion of particles in the particle system and includes: a
temperature calculation unit that calculates a temperature of a
particle, which is one of parameters of particles in the particle
system; a force calculation unit that calculates a force exerted on
the particle assumed to be immersed in a heat bath of the
temperature calculated by the temperature calculation unit; and a
state update unit that updates a state of the particle based on the
force calculated by the force calculation unit.
Inventors: |
OHNISHI; Yoshitaka;
(Yokosuka-shi, JP) ; ICHISHIMA; Daiji;
(Yokosuka-shi, JP) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
SUMITOMO HEAVY INDUSTRIES, LTD. |
Tokyo |
|
JP |
|
|
Assignee: |
SUMITOMO HEAVY INDUSTRIES,
LTD.
Tokyo
JP
|
Family ID: |
50340835 |
Appl. No.: |
14/658408 |
Filed: |
March 16, 2015 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
PCT/JP2013/005581 |
Sep 20, 2013 |
|
|
|
14658408 |
|
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Current U.S.
Class: |
703/2 |
Current CPC
Class: |
G01N 2015/0003 20130101;
G06F 2113/22 20200101; G06F 17/10 20130101; G16C 99/00 20190201;
G01N 15/10 20130101; G06F 30/20 20200101 |
International
Class: |
G06F 17/50 20060101
G06F017/50; G06F 17/10 20060101 G06F017/10; G01N 15/10 20060101
G01N015/10 |
Foreign Application Data
Date |
Code |
Application Number |
Sep 21, 2012 |
JP |
2012-208720 |
Claims
1. An analyzing device that analyzes a particle system defined in a
virtual system by numerically calculating a governing equation that
governs motion of particles in the particle system, comprising: a
temperature calculation unit that calculates a temperature of a
particle, which is one of parameters of particles in the particle
system; a force calculation unit that calculates a force exerted on
the particle assumed to be immersed in a heat bath of the
temperature calculated by the temperature calculation unit; and a
state update unit that updates a state of the particle based on the
force calculated by the force calculation unit.
2. The analyzing device according to claim 1, wherein the
temperature calculation unit includes: a Voronoi processing unit
that creates a Voronoi polyhedron in the virtual space based on a
position of a particle; and a heat conduction calculation unit that
calculates a temperature of the particle based on a heat conduction
equation discretized by using the Voronoi polyhedron created by the
Voronoi processing unit as a unit.
3. The analyzing device according to claim 2, wherein the heat
conduction equation includes a term indicating heat generation due
to deformation of the particle system.
4. The analyzing device according to claim 1, wherein material
constants are defined for a particle in the particle system so that
the particle simulates a metal particle.
5. A computer program embedded in a non-transitory computer
readable recording medium to implement on a computer a function of
analyzing a particle system defined in a virtual system by
numerically calculating a governing equation that governs motion of
particles in the particle system, the program comprising: a
temperature calculation module that calculates a temperature of a
particle, which is one of parameters of particles in the particle
system; a force calculation module that calculates a force exerted
on the particle assumed to be immersed in a heat bath of the
temperature calculated by the temperature calculation unit; and a
state updating module that updates a state of the particle based on
the force calculated by the force calculation unit.
Description
BACKGROUND OF THE INVENTION
[0001] 1. Field of the Invention
[0002] The present invention relates to an analyzing device for
analyzing a particle system.
[0003] 2. Description of the Related Art
[0004] One known method to study phenomena in material science in
general based on classical mechanics or quantum mechanics and using
a computer is simulation based on Molecular Dynamics Method
(hereinafter, MD method), Quantum Molecular Dynamics Method, or
Renormalized Molecular Dynamics (hereinafter, RMD method), which is
developed from the MD method to handle a macroscale system.
[0005] Normally, the MD method and the RMD method are only capable
of analyzing heat conduction by lattice vibration (phonons).
Therefore, the results of analysis produced by the MD method or the
RMD method in metals are often deviated from the reality because
free electrons play a great role in heat conduction.
SUMMARY OF THE INVENTION
[0006] An example of the present invention relates to an analyzing
device. The analyzing device that analyzes a particle system
defined in a virtual system by numerically calculating a governing
equation that governs motion of particles in the particle system,
and includes: a temperature calculation unit that calculates a
temperature of a particle, which is one of parameters of particles
in the particle system; a force calculation unit that calculates a
force exerted on the particle assumed to be immersed in a heat bath
of the temperature calculated by the temperature calculation unit;
and a state update unit that updates a state of the particle based
on the force calculated by the force calculation unit.
[0007] Note that any combination of the aforementioned components
or any manifestation of the present invention realized by
modifications of a method, apparatus, system, storing media,
computer program, and so forth, is effective as an example of the
present invention.
BRIEF DESCRIPTION OF THE DRAWINGS
[0008] Examples will now be described, by way of example only, with
reference to the accompanying drawings which are meant to be
exemplary, not limiting, and wherein like elements are numbered
alike in several Figures, in which:
[0009] FIG. 1 is a block diagram showing the function and
configuration of an analyzing device according to a first
example;
[0010] FIG. 2 is a data structure diagram showing an example of a
particle data storing unit;
[0011] FIG. 3 is a flowchart showing an example of a series of
steps in the analyzing device of FIG. 1;
[0012] FIG. 4 is a schematic diagram showing a particle system used
in the calculation for certification according to the first
example;
[0013] FIG. 5 is a graph showing results of calculation using the
method according to the first example;
[0014] FIG. 6 is a graph showing results of calculation not using
the method according to the first example; and
[0015] FIGS. 7A-7F are schematic diagrams showing the
time-dependent change of results of calculation obtained when the
method according to the second example is used.
DETAILED DESCRIPTION OF THE INVENTION
[0016] In the following, same reference symbols shown in Figures
indicate same or corresponding structures, parts or processes, and
redundant descriptions may be omitted.
[0017] In one related art, parameters of temperature are assigned
to particles so that a temperature field is determined by solving a
heat conduction equation using Finite Volume Method (FVM). However,
the resultant temperature field often does not reflect the
dispersion of particle velocity so that the disclosed method has a
relatively limited scope of applications.
[0018] Another related art discloses a technology of converting
kinetic energy of particles into random variables and delivering
the energy according to the Fourier's law by using the gradient of
the random variables. However, the energy may grow large depending
on the gradient of the temperature, and a particle may have a
negative energy. One measure to prevent this is to use small time
intervals. However, this will increase the computational load.
[0019] Examples of the present invention address a need to provide
an analysis technology capable of handling heat conduction
phenomena in a simulation in which a particle system including a
plurality of particles is used.
[0020] The analyzing device according to an example describes a
target of analysis, using a particle system including a plurality
of particles and analyzes the particle system by numerically
calculating the motion equation of particles. In this process, the
analyzing device determines a temperature field by solving a heat
conduction equation. The analyzing device introduces, in the motion
equation of particles, a force exerted on a particle that result
when it is assumed that the particle is immersed in heat bath of a
temperature substantially equal to the temperature as determined.
More specifically, the Langevin method (see "John C. Tully,
`Dynamics of gas-surface interaction: 3D generalized Langevin model
applied to fcc and bcc surface`, J. Chem. Phys., 1975, 73"), which
gives a random force exerted from a particle in a heat bath and a
damper resulting from viscosity, is applied. This improves physical
consistency in reproducing heat conduction of a target of analysis
such as metal, in which contribution from free electrons to heat
conduction cannot be neglected.
First Example
[0021] FIG. 1 is a block diagram showing the function and
configuration of an analyzing device 100 according to a first
example. The blocks depicted in the block diagram are implemented
in hardware such as devices or mechanical components such as a CPU
of a computer, and in software such as a computer program etc. FIG.
1 depicts functional blocks implemented by the cooperation of these
elements. Therefore, it will be obvious to those skilled in the art
having accessed this specification that the functional blocks may
be implemented in a variety of manners by a combination of hardware
and software.
[0022] In this example, although a case where the particle system
is analyzed using the MD method is described, it is obvious to
those skilled in the art having the knowledge of the specification
that the technical idea of this example can be applied to a case
where the particle system is analyzed using the RMD method or to a
case where the particle system is analyzed using particle methods
such as the Distinct Element Method (DEM), Smoothed Particle
Hydrodynamics (SPH), and Moving Particle Semi-implicit (MPS).
[0023] The analyzing device 100 is connected to an input device 102
and a display 104. The input device 102 may be a keyboard or a
mouse for receiving a user input related to a process performed in
the analyzing device 100. The input device 102 may be configured to
receive an input from a network such as the Internet or from a
recording medium such as a CD, DVD, etc.
[0024] The analyzing device 100 includes a particle system
acquisition unit 108, a temperature association unit 134, a
repeated calculation unit 120, a display control unit 118 and a
particle data storing unit 114.
[0025] The particle system acquisition unit 108 is operative to
acquire data of a particle system. The particle system comprises N
(N is a natural number) particles. Those N particles are defined in
a one, two or three dimensional virtual space, based on input
information acquired from a user through the input device 102.
Particles in the particle system may be associated with molecules
or atoms.
[0026] The particle system acquisition unit 108 is operative to
arrange N particles in the virtual space based on the input
information and to associate a velocity with each arranged
particle. In other words, the particle system acquisition unit 108
assigns an initial position and an initial velocity to the particle
system. The particle system acquisition unit 108 is operative to
associate a particle ID identifying an arranged particle and a
position of the associated particle and a velocity of the arranged
particle and to register the associated information to the particle
data storing unit 114.
[0027] The temperature association unit 134 associates a
temperature with a particle in the particle system acquired by the
particle system acquisition unit 108 based on input information
acquired from a user through the input device 102. Thus, the
temperature associated is one of the parameters of a particle. For
example, the temperature association unit 134 prompts the user via
the display 104 to input an initial value of the temperature of a
particle in the particle system. The temperature association unit
134 associates the input initial value of the temperature with the
particle ID and register the associated information in the particle
data storing unit 114.
[0028] The following descriptions assume that all particles of the
particle system are homogeneous or equivalent. The following
descriptions also assume that a potential energy function is based
on pair potential and has the same form regardless of particles.
However, it will be obvious to a person skilled in the art who has
read this specification that a technical concept according to this
example can be applied to other cases.
[0029] The repeated calculation unit 120 is operative to perform
numerical operation according to a governing equation that governs
a motion of each particle in the particle system, the particle
system being represented by data stored by the particle data
storing unit 114. In particular, the repeated calculation unit 120
is operative to perform repeated operation according to an equation
of motion of a discretized particle. The repeated calculation unit
120 includes a temperature calculation unit 110, a force
calculation unit 122, a particle state calculation unit 124, a
state update unit 126 and a termination condition deciding unit
128.
[0030] The temperature calculation unit 110 calculates the
temperature of each particle in the particle system using continuum
approximation. In particular, the temperature calculation unit 110
calculates the temperature of the particle based on a discretized
heat conduction equation. The temperature calculation unit 110
includes a Voronoi division unit 112 and a heat conduction
calculation unit 116.
[0031] The Voronoi division unit 112 performs Voronoi division of a
portion of the virtual space in which the particle system is
defined. In other words, the Voronoi division unit 112 creates
Voronoi polyhedrons in the virtual space based on the position of
the particles. First, the Voronoi division unit 112 performs
three-dimensional Delauna segmentation of the portion of the
virtual space with the particles at the vertices. Next, the Voronoi
division unit 112 creates Voronoi polyhedron elements from
tetrahedron elements obtained as a result of Delauna segmentation.
This results in Voronoi division of the portion of the virtual
space with the positions of the particles being kernel points.
[0032] The heat conduction calculation unit 116 calculates the
temperature of each particle based on a heat conduction equation
discretized by using a Voronoi polyhedron created by the Voronoi
division unit 112 as a unit. The heat conduction calculation unit
116 may analyze a temperature field using the Finite Volume Method.
For analysis of the temperature field, the heat conduction
calculation unit 116 uses the area and volume of Voronoi
polyhedrons created by the Voronoi division unit 112, and
information on the temperature field at a point of time preceding
the current time by a predetermined infinitesmal time interval
.DELTA.t (i.e., information on the temperature field of an
immediately previous cycle in the repeated calculation). In
particular, the heat conduction calculation unit 116 analyzes each
of the Voronoi polyhedrons as one control volume of FVM.
[0033] The steps of deriving a discretized heat conduction equation
used in the heat conduction calculation unit 116 will be shown
below.
[0034] The heat conduction equation is given by the following
differential equation (expression 1).
.rho. C V .differential. T .differential. t = .gradient. [ .lamda.
.gradient. T ] + Q . ( 1 ) ##EQU00001##
where .rho. denotes density, Cv denotes specific heat, T denotes
temperature, t denotes time, .lamda. denotes heat conductivity, and
Q denotes amount of heat generated per unit volume.
[0035] Taking the volume integral of both sides of expression 1, we
obtain
.rho. .intg. C V V .differential. T .differential. t V = .intg. V
.gradient. [ .lamda. .gradient. T ] V + .intg. V Q . V ( 2 )
##EQU00002##
Applying Gauss's theorem to the first term of expression 2, we
obtain
.rho. .intg. C V V .differential. T .differential. t V = .intg. S [
.lamda. .gradient. T ] S + .intg. V Q . V ##EQU00003##
[0036] Discretizing expression 3, using a Voronoi polyhedron as a
unit, we obtain
.rho. C V T i n + 1 - T i n .DELTA. t .DELTA. V i = j .DELTA. S ij
.lamda. ij T j n - T i n r ij + Q . .DELTA. V i ( 4 )
##EQU00004##
[0037] where .DELTA.V.sub.i denotes the volume of a Voronoi
polyhedron whose kernel point is located at the position of the
i-th particle, .DELTA.S.sub.ij denotes the area of the Voronoi face
between the i-th particle and the j-th particle, and r, denotes the
distance between the i-th particle and the j-th particle.
T.sub.i.sup.n denotes the temperature of the i-th particle in the
n-th cycle of the repeated calculation over time (this can be said
to be the temperature of the Voronoi polyhedron to which the i-th
particle belongs), and .lamda..sub.ij denotes the heat conductivity
between the i-th particle and the j-th particle. Expression 4 is a
discretized heat conduction equation used in the heat conduction
calculation unit 116. Expression 4 allows determining the
temperature of the i-th particle in the n+1 calculation, from the
volume and area of the Voronoi polyhedron, the temperature of the
particles determined in the n-th cycle of calculation, and the
inter-particle distance. The initial temperature associated by the
temperature association unit 134 is used as T.sub.i.sup.0.
[0038] The force calculation unit 122 calculates a force exerted on
a particle assumed to be immersed in a heat bath of a temperature
calculated by the temperature calculation unit 110. The force
calculation unit 122 includes an inter-particle action calculation
unit 130 and a heat bath action calculation unit 132.
[0039] The inter-particle action calculation unit 130 is operative
to refer to data of the particle system stored by the particle data
storing unit 114 and to calculate a force applied to each particle
in the particle system based on particle-particle distances. The
inter-particle action calculation unit 130 is operative to, with
regard to i-th (1.ltoreq.i.ltoreq.N) particle in the particle
system, identify particle(s) whose distance from the i-th particle
is less than a predetermined cut-off distance. Hereinafter, such
identified particles are called neighboring particles.
[0040] The inter-particle action calculation unit 130 is operative
to calculate a force applied to the i-th particle by each
neighboring particle based on the potential energy function between
the neighboring particle and the i-th particle and the distance
between the neighboring particle and the i-th particle. In
particular, the inter-particle action calculation unit 130 is
operative to calculate the force by obtaining a value of a gradient
of the potential energy function at the value of the distance
between the neighboring particle and the i-th particle. The
inter-particle action calculation unit 130 is operative to sum up
the force applied to the i-th particle by a neighboring particle
over all neighboring particles in order to calculate the total
force applied to the i-th particle. The force calculated by the
inter-particle action calculation unit 130 is a force based on the
interaction between particles.
[0041] Assuming that the i-th particle is immersed in a heat bath
of a substantially constant temperature at a temperature T,
calculated by the temperature calculation unit 110, the following
two forces are exerted on the i-th particle according to the
Langevin method.
[0042] (1) Damper Force (Viscous Force)
[0043] A damper force is a force exerted by the viscosity of the
heat bath on the particle. The damping constant .alpha. of the
viscous force is given by the equation below, using a Debye
frequency .omega..sub.0 and the mass m of the particle.
.alpha. = m .pi. .omega. D 6 ( 5 ) ##EQU00005##
The table below lists values of damping constant .alpha. of typical
metal substances. In this table, particles are associated with
atoms.
TABLE-US-00001 Debye Debye Damping Temper- Frequency Atomic Mass
constant Substance ature .omega..sub.D [1/s] weight m [kg] .alpha.
[kg/s] Al 428 5.60E+13 26.98 4.48173E-26 1.31E-12 Ti 420 5.50E+13
47.9 7.95681E-26 2.29E-12 Cr 630 8.25E+13 52.01 8.63953E-26
3.73E-12 Fe 470 6.15E+13 55.85 9.27741E-26 2.99E-12 Ni 450 5.89E+13
58.71 9.75249E-26 3.01E-12 Cu 343 4.49E+13 63.54 1.05548E-25
2.48E-12 Zn 327 4.28E+13 65.38 1.08605E-25 2.43E-12 Mo 450 5.89E+13
95.95 1.59385E-25 4.92E-12 Ag 225 2.95E+13 107.88 1.79203E-25
2.76E-12 W 400 5.24E+13 183.86 3.05415E-25 8.37E-12 Pt 240 3.14E+13
195.09 3.2407E-25 5.33E-12 Au 165 2.16E+13 197 3.27243E-25
3.70E-12
[0044] As shown in the table, the damping constant .alpha. is on
the order of 1.0.times.10.sup.-12 (kg/s) in case particles are
associated with metal atoms. The Debye frequency .omega..sub.D
depends on the mass of the particle. Therefore, if the mass of the
particle is .beta. times the mass of the atom, the damping constant
.alpha. will be .beta..sup.0.5 times the original. For example, if
particles that have the property of iron and that have a mass 100
times that of iron atoms are used, the damping constant .alpha.
will be 2.99.times.10.sup.-11 (kg/s).
[0045] (2) Random Force
[0046] A random force corresponds to a force produced by collision
of particles in the heat bath. The random force has a standard
deviation .sigma. given by the expression below.
.sigma. = 2 .alpha. K B T i .DELTA. t ( 6 ) ##EQU00006##
where K.sub.B denotes the Boltzman constant.
[0047] The heat bath action calculation unit 132 calculates the
viscous force and the random force exerted on the i-th
(1.ltoreq.i.ltoreq.N) particle in the particle system in accordance
with expressions 5 and 6. The heat bath action calculation unit 132
calculates the total force exerted on the i-th particle by adding
the viscous force and the random force calculated as being exerted
on the i-th particle to the force exerted on i-th particle based on
the interaction between particles. The total force F.sub.i exerted
on the i-th particle is given by expression 7 below.
F .fwdarw. i = - i .noteq. j { .differential. .phi. ij ( r ij )
.differential. r } r .fwdarw. ij r ij - .alpha. V .fwdarw. i + F
.fwdarw. random ( 7 ) ##EQU00007##
where .phi..sub.ij denotes the potential energy function between
the i-th particle and the j-th particle, v.sub.i denotes the
velocity of the i-th particle, and F.sub.random denotes the random
force having a standard deviation .sigma.. The arrow over a symbol
indicates a vector quantity.
[0048] The particle state calculation unit 124 refers to data for
the particle system stored in the particle data storing unit 114
and calculates at least one of the position and the velocity of the
particles in the particle system by applying the total force
calculated by the heat bath action calculation unit 132 to the
discretized motion equation of particles. In this example, the
particle state calculation unit 124 calculates both the position
and the velocity of the particles.
[0049] The particle state calculation unit 124 calculates the
velocity of the particles using according to the discretized motion
equation of particles that includes the total force calculated by
the heat bath action calculation unit 132. The particle state
calculation unit 124 calculates the velocity of the i-th particle
in the particle system by substituting the total force calculated
by the heat bath action calculation unit 132 as being exerted on
the i-th particle, into the motion equation of particles
discretized according to a predetermined numerical analysis method
such as the leap-frog method or the Euler's method and by using a
time interval .DELTA.t. In this calculation, the velocity of the
particle calculated in the previous cycle of repeated calculation
is used.
[0050] The particle state calculation unit 124 is operative to
calculate the position of a particle based on the calculated
velocity of the particle. The particle state calculation unit 124
is operative to calculate the position of the i-th particle of the
particle system by applying the calculated velocity of the i-th
particle to an equation of relationship between the position and
the velocity of the i-th particle, the equation being discretized
based on a certain numerical analysis method and the equation being
discretized using the ticks of time t. This calculation uses
position of the particle obtained in the previous cycle of the
repeated operation.
[0051] The state update unit 126 updates the state of each particle
in the particle system based on the result of calculation by the
particle state calculation unit 124. The state update unit 126 is
operative to update each of the position and the velocity of each
particle in the particle system stored by the particle data storing
unit 114 with the position and the velocity calculated by the
particle state calculation unit 124.
[0052] The termination condition deciding unit 128 is operative to
decide whether the repeated operation in the repeated calculation
unit 120 should be terminated or not. The termination conditions
with which the repeated operation should be terminated may include
the condition that the number of operations in the repeated
operation reaches a predetermined number, the condition that an
instruction for termination is received from outside and the
condition that the particle system reaches a steady state. The
termination condition deciding unit 128 is operative to terminate
the repeated operation in the repeated calculation unit 120 if the
termination condition is met. The termination condition deciding
unit 128 is operative to return the process to the temperature
calculation unit 110 if the termination condition is not met. Then,
the temperature calculation unit 110 is operative to again
calculate the temperature with position of particles updated by the
state update unit 126.
[0053] The display control unit 118 is operative to cause the
display 104 to display the time evolution of the particle system or
the state of the particle system at a certain time based on the
position, velocity and temperature of each particle of the particle
system, the particle system being represented by data stored by the
particle data storing unit 114. This display may be performed in a
form of still image or moving image.
[0054] FIG. 2 is a data structure diagram showing an example of the
particle data storing unit 114. The particle data storing unit 114
stores the particle ID, the position of the particle, the velocity
of the particle and the temperature of the particle.
[0055] In the above-described example, an example of the storing
unit is a hard disk or a memory. It should be understood by a
person skilled in the art who has read this specification that it
is possible to realize each unit, based on descriptions in this
specification, by a CPU (not shown), a module of installed
application program, a module of system program or a memory
temporarily storing contents of data that has been read out from a
hard disk.
[0056] A description will now be given of the operation of the
analyzing device 100 having the configuration described above. FIG.
3 is a flowchart showing an example of a series of steps in the
analyzing device 100. The analyzing device 100 determines the
initial state of the particle system, i.e., the initial position,
the initial velocity, and the initial temperature of the particles
(S202). The analyzing device 100 performs Voronoi analysis based on
the position of the particles and creates Voronoi polyhedrons
(S204). The analyzing device 100 uses FVM to analyze the
temperature field (S206) and updates the temperature of the
particles. The analyzing device 100 calculates the force exerted on
each particle based on the potential energy function between
particles (S208). The analyzing device 100 adds the viscous force
and the random force to the force calculated in step S208 (S210).
The analyzing device 100 calculates the velocity and the position
of the particles according to the motion equation of particles
including the force calculated in step S210 (S212). The analyzing
device 100 updates the position and the velocity of the particles
stored in the particle data storing unit 114 with the position and
the velocity calculated (S214). The analyzing device 100 determines
whether a termination condition is met (S216). If the termination
condition is not met (N in S216), the process is returned to step
S204. If the termination condition is met (Y in S216), the process
is terminated.
[0057] The temperature calculation unit 110 calculates the
temperature of each particle by continuum approximation. Therefore,
the temperature of the particle calculated by the temperature
calculation unit 110 may differ largely from the dispersion of
particle velocity, which is the primary definition of temperature.
In order to mitigate or remove such inconsistency, we have arrived
at an idea of determining the temperature by the temperature
calculation unit 110 and then reflecting the kinetic energy
originating from the temperature in the motion of the particle. The
velocity of the particle may be forced to be changed to the
velocity corresponding to the temperature by, for example,
temperature scaling. However, this approach places a constraint on
the motion and so is non-physical in nature.
[0058] Accordingly, the analyzing device 100 according to the
example is configured to correct the term of the force in the
motion equation based on the temperature, by assuming that the
particle is immersed in a heat bath of a temperature calculated by
the temperature calculation unit 110. This can reflect the
temperature calculated by the temperature calculation unit 110 in
the velocity field of the particles so that the temperature field
calculated by the temperature calculation unit 110 can be
introduced more naturally. This can consequently provide a model
with less physical inconsistency.
[0059] We conducted a calculation to certify the method according
to the example. Basically, the MD method, which is incorporated in
the example, is only capable of handling heat conduction by lattice
vibration of particles so that contribution from free electrons is
not reflected. Therefore, in case the MD method is used to analyze
a metal as a target, i.e., in case material constants (e.g., Debye
temperature, Debye frequency, atomic weight, and density, specific
heat, heat conductivity in the heat conduction equation) are
defined for particles in the particle system so that the particles
simulate metal particles, the method according to the example is
quite useful.
[0060] FIG. 4 is a schematic diagram showing a particle system 300
used in our calculation for certification. The particle system 300
simulates a metal bar. The temperature at the ends of the bar is
fixed to 0(K). The initial temperature distribution is given by the
following expression (8)
T ( r ) = 100 8 .pi. 2 sin { .pi. L r } ( 8 ) ##EQU00008##
where L denotes the length of the bar, r denotes the distance from
the end, and T(r) denotes the temperature at the distance r.
[0061] In this case, the theoretical formula of temperature
distribution after the elapse of time t is given by expression 9
below.
T ( r , t ) = 100 8 .pi. 2 sin ( - a .pi. 2 L 2 t ) sin ( .pi. L r
) ( 9 ) ##EQU00009##
where a denotes the thermal diffusion constant, and the
relationship given by the following expression 10 holds.
a = .lamda. .rho. C V ( 10 ) ##EQU00010##
[0062] FIG. 5 is a graph showing results of calculation using the
method according to the example. FIG. 6 is a graph showing results
of calculation not using the method according to the example.
According to the method of the example, the calculated value of
temperature distribution (denoted by solid dots) agrees well with
the theoretical value (denoted by the solid line) after the elapse
of 0.3 (ns), 0.6 (ns), and 0.9 (ns) since the time evolution of the
particle system 300 is started. This is in contrast with the case
of the ordinary MD method that does not incorporate the method
according to the example, where heat diffusion is extremely slower
as compared with theoretical values.
Second Example
[0063] In the second example, a description is given of a case
where variation in the structure of the particle system, i.e., heat
generation associated with variation in the particle arrangement,
is considered. The analyzing device 100 according to the second
example has the same configuration as that of FIG. 1. The following
description focuses on the difference from the first example.
[0064] As in the first example, the heat conduction calculation
unit 116 calculates the temperature of each particle according to a
heat conduction equation discretized by using a Voronoi polyhedron
created by the Voronoi division unit 112 as a unit. As in the first
example, the discretized heat conduction equation used in the heat
conduction calculation unit 116 is given by expression 4. In this
example, heat generation associated with variation in the structure
of the particle system is considered so that the amount of heat
generated Q of expression 4 is given by the following expression
11.
Q . = [ i .noteq. j .phi. ( r ij ) + K i ] n - [ i .noteq. j .phi.
( r ij ) + K i ] n - 1 .DELTA. t .DELTA. V i + i .noteq. j F rij v
rij .DELTA. V i ( 11 ) ##EQU00011##
where .phi. denotes the potential energy function between
particles, K.sub.i denotes the kinetic energy of the i-th particle,
n denotes the number of times of repeated calculation with time,
.DELTA.t denotes the time interval, F.sub.rij denotes the friction
created between the i-th particle and the j-th particle, and
v.sub.rij denotes the relative velocity between the i-th particle
and the j-th particle. Expression 11 indicates that heat is
generated if the total energy of the particles is increased due to
deformation of the particle system. Expressions 4 and 11 determine
the temperature of the i-th particle in the n+1-th calculation.
[0065] As in the first example, the analyzing device 100 according
to this example is capable of reflecting the temperature calculated
by the temperature calculation unit 110 in the temperature field of
the particles so that the temperature field calculated by the
temperature calculation unit 110 can be introduced more naturally.
This can consequently provide a model with less physical
inconsistency. In further accordance with the analyzing device 100
of this example, heat generation associated with variation in the
structure of the particle system can be reflected in the
temperature calculated by the temperature calculation unit 110
(heat conduction calculation unit 116). Accordingly, there is
provided a model with less physical inconsistency in the presence
of deformation in the structure of the particle system. This allows
more accurate simulation of a phenomenon in which heat generation
associated with deformation of a metal such as plastic forming is
involved and allows prediction of temperature increase during
work.
[0066] We conducted a calculation to certify the method according
to this example. FIGS. 7A-7F are schematic diagrams showing results
of calculation obtained when the method according to this example
is used. A particle system 400 simulates a metal block of 0.6 mm (X
direction).times.0.6 mm (Y direction).times.0.95 mm (Z direction).
FIGS. 7A-7F show the time-dependent change occurring when a pull
force equivalent to 50 GPa is exerted on one upper layer and one
lower layer of the particle system 400. FIG. 7A shows the moment
when the pull force is exerted on the particle system 400. FIGS.
7B, 7C, 7D, 7E, and 7F show the state occurring after the elapse of
25 .mu.s, 50 .mu.s, 75 .mu.s, 100 .mu.s, and 125 .mu.s since the
time evolution of the particle system 400 is started. The results
show that the temperature of the particle system 400 is increased
as a result of the exertion of the pull force and the associated
variation in the structure of the particle system 400. This is in
agreement with the knowledge that variation in the structure of the
particle system 400 generates heat.
[0067] The structure and operation of the analyzing device 100
according to the examples are described above. The examples are
intended to be illustrative only and it will be obvious to those
skilled in the art that various modifications to combinations of
constituting elements and processes could be developed and that
such modifications are also within the scope of the present
invention.
[0068] The repeated calculation unit 120 according to the examples
are described as calculating both the position and velocity of the
particle. However, the description is non-limiting as to the mode
of calculation. For example, some numerical analysis methods like
the Verlet method directly calculate the position of a particle by
referring to the force exerted on the particle and so do not
require positively calculating the velocity of the particle. The
technical idea according to the examples may also be applied to
such methods.
[0069] Priority is claimed to Japanese Patent Application No.
2012-208720, filed Sep. 21, 2012, and International Patent
Application No. PCT/JP2013/005581, the entire content of each of
which is incorporated herein by reference.
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